OFFSET
0,3
COMMENTS
a(n) is also the number of single-elimination sports tournament schedules possible for 2n+1 teams if matches involve three teams, arbitrarily many arenas are available, and labeled teams have been specified, but the bracket of matches has not been specified.
LINKS
Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307.
FORMULA
a(n) = Y(2n+1), where Y(n) = Sum_{i=1..floor(n/3)} (n!/(i!*6^i*(n-3*i)!))*Y(n-2*i), with Y(1)=1.
EXAMPLE
Consider 7 named players in a sport in which players compete 3 at a time (e.g. the television gameshow "Jeopardy!"). The number of ways a single-elimination tournament can be arranged, if simultaneous matches can take place, is a(3)=420. Three of these 420 are: (1) A, B, and C play; the winner plays against D and E; the winner plays against F and G. (2) D, E, and F play; the winner plays against A and B; the winner plays against C and G. (3) A, B, and C play simultaneous with D, E, and F; the winners of these matches play against G.
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add((2*n+1)!/
(i!*6^i*(2*n+1-3*i)!)*a(n-i), i=1..(2*n+1)/3))
end:
seq(a(n), n=0..15); # Alois P. Heinz, Feb 25 2025
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Noah A Rosenberg, Feb 25 2025
STATUS
approved